Secion 2.3 Linear and Angular Velociies The mos inuiive measure of he rae a which he rider is raveling around he wheel is wha we call linear velociy. Anoher way o specify how fas he rider is raveling around he wheel is wih wha we call angular velociy. Linear Speed Definiion If P is a poin on a circle of ius r, and P moves a disance s on he circumference of he circle in an amoun of ime, hen he linear velociy, v, of P is given by he formula speed disance ime v s Example A poin on a circle ravels 5 cm in 2 sec. Find he linear velociy of he poin. Given: s = 5cm = 2 sec v s 5 cm 2 sec 2.5 cm / sec 21
Angular Speed Definiion If P is a poin moving wih uniform circular moion on a circle of ius r, and he line from he cener of he circle hrough P sweeps ou a cenral angle in an amoun of ime, hen he angular velociy, (omega), of P is given by he formula where is measured in ians Example A poin on a circle roaes hrough 3 4 ians in 3 sec. Give he angular velociy of he poin. Given: = 3 4 = 3sec 3 4 3 sec / sec 4 Example A bicycle wheel wih a ius of 13.0 in. urns wih an angular velociy of 3 ians per seconds. Find he disance raveled by a poin on he bicycle ire in 1 minue. Given: r = 13.0 in. = 3/sec = 1 min = 60 sec. s r s r s r s r 360 13 2,340 inches or 2,340 12 195 f 22
Relaionship beween he Two Velociies If s r s r s r v r v r Linear and Angular Velociy If a poin is moving wih uniform circular moion on a circle of ius r, hen he linear velociy v and angular velociy of he poin are relaed by he formula v r Example A phonograph record is urning a 45 revoluions per minue (rpm). If he disance from he cener of he record o a poin on he edge of he record is 3 inches, find he angular velociy and he linear velociy of he poin in fee per minue. 45 rpm v 45 rev min 45 rev 2 min 1 rev 90 / min r (3 in.) 90 min 270 in min 848 in / min 1 f v 848 in min 12in v 70.7 f / min 1 revoluion = 2 23
Example Suppose ha P is on a circle wih ius 10 cm, and ray OP is roaing wih angular speed a) Find he angle generaed by P in 6 seconds b) Find he disance raveled by P along he circle in 6 seconds. c) Find he linear speed of P in cm per sec. a).6 18 3 / sec. 18 b) s r 10 s 10 cm 3 3 c) v s 10 v 3 6 10 18 5 cm / se 9 c Example A bel runs a pulley of ius 6 cm a 80 rev / min. a) Find he angular speed of he pulley in ians per sec. b) Find he linear speed of he bel in cm per sec. a) 80 rev 1min 2 min 60sec 1 rev b) v r 8 / se 3 c 6 8 3 50 cm / sec 24
Example The diameer of he Ferris wheel is 250 f, he disance from he ground o he boom of he wheel is 14 f, and one complee revoluion akes 20 minues, find a. The linear velociy, in miles per hour, of a person riding on he wheel. b. The heigh of he rider in erms of he ime, where is measured in minues. Given: = 1 rev= 2 = 20 min. r D 250 125 f 2 2 a. 2 20 or v r 10 / min v r (125 f) 39.27 f 10 / min / min v 39.27 0.45 mi / hr f 60min 1mile min 1hr 5,280 f b. cos OP OP 1 OP 125 OP 125cos H PP 14 0 OP OP 14 0 125 125cos 14 139 125cos P 14 f P 0 O P 1 H 10 H 139 125cos 10 25
s Secion 2.3 Linear and Angular Velociies 1. Find he linear velociy of a poin moving wih uniform circular moion, if s = 12 cm and = 2 sec. 2. Find he disance s covered by a poin moving wih linear velociy v = 55 mi/hr and = 0.5 hr. 3. Poin P sweeps ou cenral angle = 12 as i roaes on a circle of ius r wih = 5 sec. Find he angular velociy of poin P. 4. Find an equaion ha expresses l in erms of ime. Find l when is 0.5 sec, 1.0 sec, and 1.5 sec. (assume he ligh goes hrough one roaion every 4 seconds.) 5. Find he angular velociy, in ians per minue, associaed wih given 7.2 rpm. 6. When Lance Armsrong blazed up Moun Venoux in he 2002 our, he was equipped wih a 150- millimeer-diameer chainring and a 95-millimeer-diameer sprocke. Lance is known for mainaining a very high cadence, or pedal rae. The sprocke and rear wheel roae a he same rae, and he diameer of he rear wheel is 700 mm. If he was pedaling a a rae of 90 revoluions per minue, find his speed in kilomeers per hour. (1 km = 1,000,000 mm or 10 6 mm) 26
7. A Ferris wheel has a ius 50.0 f. A person akes a sea and hen he wheel urns 2 3. a) How far is he person above he ground? b) If i akes 30 sec for he wheel o urn 2 3, wha is he angular speed of he wheel? 8. A fire ruck parked on he shoulder of a freeway nex o a long block wall. The red ligh on he op of he ruck is 10 fee from he wall and roaes hrough a complee revoluion every 2 seconds. Find he equaions ha give he lenghs d and in erms of ime. 9. Suppose ha poin P is on a circle wih ius 60 cm, and ray OP is roaing wih angular speed 12 ian per sec. a) Find he angle generaed by P in 8 sec. b) Find he disance raveled by P along he circle in 8 sec. c) Find he linear speed of P in 8 sec. 10. Tires of a bicycle have ius 13 in. and are urning a he rae of 215 revoluions per min. How fas is he bicycle raveling in miles per hour? (Hin: 1 mi = 5280 f.) 27
11. Earh ravels abou he sun in an orbi ha is almos circular. Assume ha he orbi is a circle wih ius 93,000,000 mi. Is angular and linear speeds are used in designing solar-power faciliies. a) Assume ha a year is 365 days, and find he angle formed by Earh s movemen in one day. b) Give he angular speed in ians per hour. c) Find he linear speed of Earh in miles per hour. 12. Earh revolves on is axis once every 24 hr. Assuming ha earh s ius is 6400 km, find he following. a) Angular speed of Earh in ians per day and ians per hour. b) Linear speed a he Norh Pole or Souh Pole c) Linear speed ar a ciy on he equaor 13. The pulley has a ius of 12.96 cm. Suppose i akes 18 sec for 56 cm of bel o go around he pulley. a) Find he linear speed of he bel in cm per sec. b) Find he angular speed of he pulley in per sec. 14. The wo pulleys have ii of 15 cm and 8 cm, respecively. The larger pulley roaes 25 imes in 36 sec. Find he angular speed of each pulley in per sec. 15. A hread is being pulled off a spool a he rae of 59.4 cm per sec. Find he ius of he spool if i makes 152 revoluions per min. 28
16. A railroad rack is laid along he arc of a circle of ius 1800 f. The circular par of he rack subends a cenral angle of 40. How long (in seconds) will i ake a poin on he fron of a rain raveling 30 mph o go around his porion of he rack? 17. A 90-horsepower ouboard moor a full hrole will roae i propeller a exacly 5000 revoluions per min. Find he angular speed of he propeller in ians per second. 18. The shoulder join can roae a 25 /min. If a golfer s arm is sraigh and he disance from he shoulder o he club head is 5.00 f, find he linear speed of he club head from he shoulder roaion 29
Secion 2.3 Linear and Angular Velociies Find he linear velociy of a poin moving wih uniform circular moion, if s = 12 cm and = 2 sec. v s 12 cm 2 sec 6 cm / sc e Find he disance s covered by a poin moving wih linear velociy v = 55 mi/hr and = 0.5 hr. s v 55 mi 0.5 hr hr 27.5 miles Poin P sweeps ou cenral angle = 12 as i roaes on a circle of ius r wih = 5 sec. Find he angular velociy of poin P. 12 5 sec 2.4 / sec 17
Find an equaion ha expresses l in erms of ime. Find l when is 0.5 sec, 1.0 sec, and 1.5 sec. (assume he ligh goes hrough one roaion every 4 seconds.) 2 / sec 4 sec 2 / sec 2 2 cos 100 2 l l cos 100 2 l 100 100sec 2 cos 2 For = 0.5 sec l 100 100 100 100 2 141 f cos 1 1 cos 2 2 4 2 For = 1.0 sec l 100 100 Undefined cos 0 2 100 100 100 cos 3 3 1 cos 2 2 4 2 For = 1.5 sec l 100 2 141 f Find he angular velociy, in ians per minue, associaed wih given 7.2 rpm. 7.2 rev 2 ians 14.4 45.2 min rev min 18
When Lance Armsrong blazed up Moun Venoux in he 2002 our, he was equipped wih a 150- millimeer-diameer chainring and a 95-millimeer-diameer sprocke. Lance is known for mainaining a very high cadence, or pedal rae. The sprocke and rear wheel roae a he same rae, and he diameer of he rear wheel is 700 mm. If he was pedaling a a rae of 90 revoluions per minue, find his speed in kilomeers per hour. (1 km = 1,000,000 mm or 10 6 mm) Chainring: v r 90 rev 2 ians 60 min min rev 1 hr v 10800 hr r 150 ( mm ) 10800 2 hr 810000 mm hr Sprocke: v r 810000mm hr 95mm 2 17052.63 hr vr 350( mm) 1 km 17052.63 10 6 mm hr 18.8 km hr 19
A fire ruck parked on he shoulder of a freeway nex o a long block wall. The red ligh on he op of he ruck is 10 fee from he wall and roaes hrough a complee revoluion every 2 seconds. Find he equaions ha give he lenghs d and in erms of ime. 2 2 / sec an 10 d d 10an 10 an sec 10 l l 10sec 10sec Suppose ha poin P is on a circle wih ius 60 cm, and ray OP is roaing wih angular speed 12 ian per sec. a) Find he angle generaed by P in 8 sec. b) Find he disance raveled by P along he circle in 8 sec. c) Find he linear speed of P in 8 sec. a).8 2 12 3 b) s r s 2 60 40 cm 3 c) v s v 40 5 cm / sec 8 20
A Ferris wheel has a ius 50.0 f. A person akes a sea and hen he wheel urns 2 3. a) How far is he person above he ground? b) If i akes 30 sec for he wheel o urn 2 3, wha is he angular speed of he wheel? a) 2 3 2 6 h cos 1 r h rcos 1 h 50cos 43.3 f 1 6 Person is 50 43.3 93.3 f above he ground b) 2 3 30 sec / sec 45 Tires of a bicycle have ius 13 in. and are urning a he rae of 215 revoluions per min. How fas is he bicycle raveling in miles per hour? (Hin: 1 mi = 5280 f.) 215 rev 2 430 / min 1 rev / min v r 13 430 5590 in 60 min 1f v 5590 in 1mi min 1hr 12in 5280 f 16.6 mph 21
Earh ravels abou he sun in an orbi ha is almos circular. Assume ha he orbi is a circle wih ius 93,000,000 mi. Is angular and linear speeds are used in designing solar-power faciliies. a) Assume ha a year is 365 days, and find he angle formed by Earh s movemen in one day. b) Give he angular speed in ians per hour. c) Find he linear speed of Earh in miles per hour. 2 365 365 a) 1 2 1 b) 2 day 365 days 24 hr 4380 / hr vr 67,000 mph 4380 c) 93,000,000 Earh revolves on is axis once every 24 hr. Assuming ha earh s ius is 6400 km, find he following. a) Angular speed of Earh in ians per day and ians per hour. b) Linear speed a he Norh Pole or Souh Pole c) Linear speed ar a ciy on he equaor a) 2 1 day 2 1 day / hr 1 day 24 hr 12 b) A he poles, r = 0 so v = rw = 0 c) A he equaor, r = 6400 km v rw 6400 2 12,800 km / day 1 12,800 km day day 24 hr 533 km / hr 22
The pulley has a ius of 12.96 cm. Suppose i akes 18 sec for 56 cm of bel o go around he pulley. a) Find he linear speed of he bel in cm per sec. b) Find he angular speed of he pulley in per sec. Given: s = 56 cm in = 18 sec r = 12.96 cm a) v s 56 18 3.1 cm / sec b) v 3.1.24 / sec r 12.96 The wo pulleys have ii of 15 cm and 8 cm, respecively. The larger pulley roaes 25 imes in 36 sec. Find he angular speed of each pulley in per sec. Given: 25 / sec 36 imes r 15 cm r 8 cm 1 2 The angular velociy of he larger pulley is: 25 imes 2 36 sec 1 ime 25 rd a / sec 18 The linear velociy of he larger pulley is: 25 cm / sec vr 15 125 18 6 The angular velociy of he smaller pulley is: v 1 v r r 1125 8 6 125 48 / sec 23
A hread is being pulled off a spool a he rae of 59.4 cm per sec. Find he ius of he spool if i makes 152 revoluions per min. Given: 152 rev / min v 59.4 cm / sec r v 1 v 1 59.4 cm 152 rev sec min 1 min 60 sec 1 rev 59.4 cm 152 3.7 cm rev 1 min 2 sec A railroad rack is laid along he arc of a circle of ius 1800 f. The circular par of he rack subends a cenral angle of 40. How long (in seconds) will i ake a poin on he fron of a rain raveling 30 mph o go around his porion of he rack? Given: r = 1800 f. 40 40 2 180 9 v 30 mph The arc lengh: 2 v s s v 400 f 30 mi hr 40 f hr 1mi 3600sec 3 mi 5280 f 1hr 29 sec s r 1800 400 f 9 24
A 90-horsepower ouboard moor a full hrole will roae i propeller a exacly 5000 revoluions per min. Find he angular speed of he propeller in ians per second. 5000 rev 2 1 min min 1 rev 60 sec 523.6 / sec The shoulder join can roae a 25 /min. If a golfer s arm is sraigh and he disance from he shoulder o he club head is 5.00 f, find he linear speed of he club head from he shoulder roaion. Given: 25 / min r 5 f 125 / min v r 5 25 f 25